Every planar graph with the Liouville property is amenable

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic functions proved by Benjamini and Schramm. As a corollary we obtain that every planar non-amenable graph admits Dirichlet harmonic functions

Mixing time of critical Ising model on trees is polynomial in the height

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature $\beta_c$, the inverse-gap is bounded for $\beta < \beta_c$, polynomial in the surface area for $\beta = \beta_c$ and exponential in it for $\beta > \beta_c$. This has been proved for $\Z^2$ except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for $\beta \beta_c$ were established, where $\beta_c$ is the critical spin-glass parameter, and the tree-height $h$ plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the $b$-ary tree, by showing that it is indeed polynomial in $h$ at criticality. The degree of our polynomial bound does not depend on $b$, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for $\beta > \beta_c$, the inverse-gap and mixing-time are both $\exp[\Theta((\beta-\beta_c) h)]$.Comment: 53 pages; 3 figure

Multivariate tight affine frames with a small number of generators

We give a simple and explicit construction of compactly supported affine tight frames with small number of generators, associated to multivariate box splines (with respect to the dilation matrix 2I). Moreover, the same technique applied to the case of bivariate box splines on the four-directions mesh with dilation matrix gives tight frames with at most five generators

Multivariate compactly supported biorthogonal spline wavelets

We study biorthogonal bases of compactly supported wavelets constructed from box splines in \uc2N with any integer dilation factor. For a suitable class of box splines we write explicitly dual low-pass filters of arbitrarily high regularity and indicate how to construct the corresponding high-pass filters (primal and dual)